The "white" and "rye" sounding similar suggestion works, but I figure he may have thought his question sounded more like "Just gimme a cheese sandwich, Swiss cheese, alright?". It kinda sounds like it, especially when the waitress repeats it.

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All that plus being a businessman, he probably has to do things more efficient with his diction and spelling. Probably a part of who he is

I was wondering the same thing, but I think the poster who said rye and white can sometimes be confused may be right. Or he just could have been so nervous that he didn't really know what he was saying.
R.I.P. Rick Ducommun and Tony Longo

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Yeah! I guess! And a cheese sandwich on rye? OK? Doesn't sound all that good? Not even a grilled cheese sandwich, but a plain cold cheese sandwich?

His life, and the current situation, were out of control. Being particular with things was a coping mechanism- a stab at stabilization.

He's an arrogant ass that needed a good bitch slap. Too bad that big guy didn't beat him up more.

'Twas a ponderous moment.

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Rye bread was more popular back in the day than it is now! But back east more, not so much in California ever!
Just another odd thing about this movie. Strange not to order a meat sandwich and on an unusual bread type too!

b/c he's a s-p-a-z?
'I promise you a police car on every sidewalk.'- Marion Barry

I'm surprised they even had rye in the diner. He didn't even eat it anyway, I didn't think he intended to.

wheel of fortune is his favorite tv show.

That was funny nuts
R.I.P. Rick Ducommun and Tony Longo

he does seem like a huge fan of sushi. they way he pecks at his food.
The food I've liked in my time is American country cookin'-Colonel Sanders

spelling outs-o-u-r-d-o-u-g-hwould have been impressive.
Too bad they didn't have that spicy green horseradish sauce back then.
I could see myself laughing at him saying "That's w-a-s-a-b-i!"
I'd say this cloud is Cumulo Nimbus.
Didn't he discover America?
Penfold, shush.

spelling outs-o-u-r-d-o-u-g-hwould have been impressive.

I love everything about the diner segment. I love the backwards camera-walk to the bathroom. I love his inner-dialog with himself about all that's happened thus far. I love the guy that stuffs his mouth full of sandwich bites and then sips his beer. And I always thought about Mann spelling out RYE for the simple reason someone else sort of mentioned - trying to seperate himself from simple townfolk - Mr Mann seemed a little arrogant.
I love this movie so much, and I swear I think about it and the whole diner scene whenever I eat a sandwich.

Well, can you blame him? If you'd gone through a fright, or a nerve-wracking, harrowing experience, I think you'd be a little shaken and nervous too, and not exactly rational.

Remember how nervous he was in Touch Of Evil? lol
"I've seen things in this city that make Dante's Inferno read like Winnie The Pooh."

b/c he's a j-a-c-k-a-s-s?
His namewas Julio Iglesias!

I had a feeling Dennis Weaver might have been a vegetarian so I looked it up. Yes, he was. Starting in 1958.
"40 years ago on the set of Gunsmoke I read the book The Holy Science. Since then I have not eaten meat."

It wouldn't surprise me at all if Mann was a vegetarian.